LMFDB - Embedded newform 160.8.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [160,8,Mod(1,160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("160.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,8,0,250,0,1288,0,-3974] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.9816040775$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
Character	$\chi$	$=$	160.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+17.5647 q^{3} +125.000 q^{5} +1661.35 q^{7} -1878.48 q^{9} -5622.79 q^{11} -11031.4 q^{13} +2195.58 q^{15} -16752.0 q^{17} -35859.5 q^{19} +29181.0 q^{21} +51497.4 q^{23} +15625.0 q^{25} -71408.8 q^{27} +197277. q^{29} -19398.8 q^{31} -98762.3 q^{33} +207669. q^{35} -576835. q^{37} -193763. q^{39} -537158. q^{41} -5819.80 q^{43} -234810. q^{45} -1.04266e6 q^{47} +1.93654e6 q^{49} -294243. q^{51} +466618. q^{53} -702848. q^{55} -629859. q^{57} -498114. q^{59} +2.26280e6 q^{61} -3.12082e6 q^{63} -1.37893e6 q^{65} -1.93123e6 q^{67} +904534. q^{69} +2.41337e6 q^{71} +2.63563e6 q^{73} +274448. q^{75} -9.34141e6 q^{77} -8.41287e6 q^{79} +2.85397e6 q^{81} -5.51975e6 q^{83} -2.09400e6 q^{85} +3.46510e6 q^{87} -5.97135e6 q^{89} -1.83270e7 q^{91} -340734. q^{93} -4.48243e6 q^{95} +1.64459e7 q^{97} +1.05623e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{3} + 250 q^{5} + 1288 q^{7} - 3974 q^{9} - 2944 q^{11} + 2028 q^{13} + 1000 q^{15} - 27644 q^{17} - 53488 q^{19} + 32752 q^{21} - 42120 q^{23} + 31250 q^{25} - 30448 q^{27} + 143228 q^{29} + 255664 q^{31}+ \cdots + 4948864 q^{99}+O(q^{100}) $ 2 * q + 8 * q^3 + 250 * q^5 + 1288 * q^7 - 3974 * q^9 - 2944 * q^11 + 2028 * q^13 + 1000 * q^15 - 27644 * q^17 - 53488 * q^19 + 32752 * q^21 - 42120 * q^23 + 31250 * q^25 - 30448 * q^27 + 143228 * q^29 + 255664 * q^31 - 124384 * q^33 + 161000 * q^35 - 324164 * q^37 - 318672 * q^39 - 257180 * q^41 - 594296 * q^43 - 496750 * q^45 - 2160920 * q^47 + 1252386 * q^49 - 190064 * q^51 + 849244 * q^53 - 368000 * q^55 - 461248 * q^57 - 2244720 * q^59 + 116332 * q^61 - 2338456 * q^63 + 253500 * q^65 - 4929384 * q^67 + 1799952 * q^69 - 3456432 * q^71 + 1393940 * q^73 + 125000 * q^75 - 10341536 * q^77 - 11340512 * q^79 + 7045090 * q^81 - 7882552 * q^83 - 3455500 * q^85 + 3982064 * q^87 - 6735180 * q^89 - 23202768 * q^91 - 2971616 * q^93 - 6686000 * q^95 + 23786180 * q^97 + 4948864 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	17.5647	0.375591	0.187796	−	0.982208i	$-0.439866\pi$
$3$	17.5647	0.375591	0.187796	+	0.982208i	$0.439866\pi$
$4$	0	0
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	1661.35	1.83070	0.915351	−	0.402656i	$-0.131913\pi$
$7$	1661.35	1.83070	0.915351	+	0.402656i	$0.131913\pi$
$8$	0	0
$9$	−1878.48	−0.858931
$10$	0	0
$11$	−5622.79	−1.27373	−0.636865	−	0.770976i	$-0.719769\pi$
$11$	−5622.79	−1.27373	−0.636865	+	0.770976i	$0.719769\pi$
$12$	0	0
$13$	−11031.4	−1.39261	−0.696305	−	0.717746i	$-0.745174\pi$
$13$	−11031.4	−1.39261	−0.696305	+	0.717746i	$0.745174\pi$
$14$	0	0
$15$	2195.58	0.167969
$16$	0	0
$17$	−16752.0	−0.826979	−0.413490	−	0.910509i	$-0.635690\pi$
$17$	−16752.0	−0.826979	−0.413490	+	0.910509i	$0.635690\pi$
$18$	0	0
$19$	−35859.5	−1.19941	−0.599703	−	0.800223i	$-0.704715\pi$
$19$	−35859.5	−1.19941	−0.599703	+	0.800223i	$0.704715\pi$
$20$	0	0
$21$	29181.0	0.687596
$22$	0	0
$23$	51497.4	0.882546	0.441273	−	0.897373i	$-0.354527\pi$
$23$	51497.4	0.882546	0.441273	+	0.897373i	$0.354527\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−71408.8	−0.698198
$28$	0	0
$29$	197277.	1.50205	0.751023	−	0.660276i	$-0.229561\pi$
$29$	197277.	1.50205	0.751023	+	0.660276i	$0.229561\pi$
$30$	0	0
$31$	−19398.8	−0.116952	−0.0584762	−	0.998289i	$-0.518624\pi$
$31$	−19398.8	−0.116952	−0.0584762	+	0.998289i	$0.518624\pi$
$32$	0	0
$33$	−98762.3	−0.478401
$34$	0	0
$35$	207669.	0.818715
$36$	0	0
$37$	−576835.	−1.87217	−0.936086	−	0.351772i	$-0.885579\pi$
$37$	−576835.	−1.87217	−0.936086	+	0.351772i	$0.885579\pi$
$38$	0	0
$39$	−193763.	−0.523052
$40$	0	0
$41$	−537158.	−1.21719	−0.608595	−	0.793481i	$-0.708266\pi$
$41$	−537158.	−1.21719	−0.608595	+	0.793481i	$0.708266\pi$
$42$	0	0
$43$	−5819.80	−0.0111627	−0.00558134	−	0.999984i	$-0.501777\pi$
$43$	−5819.80	−0.0111627	−0.00558134	+	0.999984i	$0.501777\pi$
$44$	0	0
$45$	−234810.	−0.384126
$46$	0	0
$47$	−1.04266e6	−1.46487	−0.732434	−	0.680838i	$-0.761616\pi$
$47$	−1.04266e6	−1.46487	−0.732434	+	0.680838i	$0.761616\pi$
$48$	0	0
$49$	1.93654e6	2.35147
$50$	0	0
$51$	−294243.	−0.310606
$52$	0	0
$53$	466618.	0.430523	0.215261	−	0.976556i	$-0.430940\pi$
$53$	466618.	0.430523	0.215261	+	0.976556i	$0.430940\pi$
$54$	0	0
$55$	−702848.	−0.569629
$56$	0	0
$57$	−629859.	−0.450486
$58$	0	0
$59$	−498114.	−0.315753	−0.157876	−	0.987459i	$-0.550465\pi$
$59$	−498114.	−0.315753	−0.157876	+	0.987459i	$0.550465\pi$
$60$	0	0
$61$	2.26280e6	1.27642	0.638208	−	0.769864i	$-0.279676\pi$
$61$	2.26280e6	1.27642	0.638208	+	0.769864i	$0.279676\pi$
$62$	0	0
$63$	−3.12082e6	−1.57245
$64$	0	0
$65$	−1.37893e6	−0.622794
$66$	0	0
$67$	−1.93123e6	−0.784464	−0.392232	−	0.919866i	$-0.628297\pi$
$67$	−1.93123e6	−0.784464	−0.392232	+	0.919866i	$0.628297\pi$
$68$	0	0
$69$	904534.	0.331477
$70$	0	0
$71$	2.41337e6	0.800240	0.400120	−	0.916463i	$-0.368968\pi$
$71$	2.41337e6	0.800240	0.400120	+	0.916463i	$0.368968\pi$
$72$	0	0
$73$	2.63563e6	0.792966	0.396483	−	0.918042i	$-0.370231\pi$
$73$	2.63563e6	0.792966	0.396483	+	0.918042i	$0.370231\pi$
$74$	0	0
$75$	274448.	0.0751182
$76$	0	0
$77$	−9.34141e6	−2.33182
$78$	0	0
$79$	−8.41287e6	−1.91977	−0.959885	−	0.280394i	$-0.909535\pi$
$79$	−8.41287e6	−1.91977	−0.959885	+	0.280394i	$0.909535\pi$
$80$	0	0
$81$	2.85397e6	0.596694
$82$	0	0
$83$	−5.51975e6	−1.05961	−0.529806	−	0.848119i	$-0.677735\pi$
$83$	−5.51975e6	−1.05961	−0.529806	+	0.848119i	$0.677735\pi$
$84$	0	0
$85$	−2.09400e6	−0.369836
$86$	0	0
$87$	3.46510e6	0.564156
$88$	0	0
$89$	−5.97135e6	−0.897858	−0.448929	−	0.893567i	$-0.648194\pi$
$89$	−5.97135e6	−0.897858	−0.448929	+	0.893567i	$0.648194\pi$
$90$	0	0
$91$	−1.83270e7	−2.54946
$92$	0	0
$93$	−340734.	−0.0439263
$94$	0	0
$95$	−4.48243e6	−0.536390
$96$	0	0
$97$	1.64459e7	1.82961	0.914803	−	0.403901i	$-0.132346\pi$
$97$	1.64459e7	1.82961	0.914803	+	0.403901i	$0.132346\pi$
$98$	0	0
$99$	1.05623e7	1.09405
$100$	0	0
$101$	−6.90100e6	−0.666480	−0.333240	−	0.942842i	$-0.608142\pi$
$101$	−6.90100e6	−0.666480	−0.333240	+	0.942842i	$0.608142\pi$
$102$	0	0
$103$	8.15511e6	0.735359	0.367680	−	0.929953i	$-0.380152\pi$
$103$	8.15511e6	0.735359	0.367680	+	0.929953i	$0.380152\pi$
$104$	0	0
$105$	3.64763e6	0.307502
$106$	0	0
$107$	−1.19056e7	−0.939521	−0.469761	−	0.882794i	$-0.655660\pi$
$107$	−1.19056e7	−0.939521	−0.469761	+	0.882794i	$0.655660\pi$
$108$	0	0
$109$	−2.64454e6	−0.195595	−0.0977973	−	0.995206i	$-0.531180\pi$
$109$	−2.64454e6	−0.195595	−0.0977973	+	0.995206i	$0.531180\pi$
$110$	0	0
$111$	−1.01319e7	−0.703171
$112$	0	0
$113$	−1.49698e7	−0.975979	−0.487990	−	0.872849i	$-0.662270\pi$
$113$	−1.49698e7	−0.975979	−0.487990	+	0.872849i	$0.662270\pi$
$114$	0	0
$115$	6.43717e6	0.394687
$116$	0	0
$117$	2.07223e7	1.19616
$118$	0	0
$119$	−2.78309e7	−1.51395
$120$	0	0
$121$	1.21286e7	0.622386
$122$	0	0
$123$	−9.43499e6	−0.457166
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−1.02177e7	−0.442628	−0.221314	−	0.975203i	$-0.571035\pi$
$127$	−1.02177e7	−0.442628	−0.221314	+	0.975203i	$0.571035\pi$
$128$	0	0
$129$	−102223.	−0.00419260
$130$	0	0
$131$	1.19319e7	0.463726	0.231863	−	0.972748i	$-0.425518\pi$
$131$	1.19319e7	0.463726	0.231863	+	0.972748i	$0.425518\pi$
$132$	0	0
$133$	−5.95751e7	−2.19576
$134$	0	0
$135$	−8.92610e6	−0.312244
$136$	0	0
$137$	2.15335e7	0.715474	0.357737	−	0.933822i	$-0.383549\pi$
$137$	2.15335e7	0.715474	0.357737	+	0.933822i	$0.383549\pi$
$138$	0	0
$139$	1.92850e7	0.609072	0.304536	−	0.952501i	$-0.401499\pi$
$139$	1.92850e7	0.609072	0.304536	+	0.952501i	$0.401499\pi$
$140$	0	0
$141$	−1.83139e7	−0.550191
$142$	0	0
$143$	6.20273e7	1.77381
$144$	0	0
$145$	2.46596e7	0.671736
$146$	0	0
$147$	3.40147e7	0.883192
$148$	0	0
$149$	−2.96727e7	−0.734860	−0.367430	−	0.930051i	$-0.619762\pi$
$149$	−2.96727e7	−0.734860	−0.367430	+	0.930051i	$0.619762\pi$
$150$	0	0
$151$	6.94126e7	1.64066	0.820331	−	0.571890i	$-0.193789\pi$
$151$	6.94126e7	1.64066	0.820331	+	0.571890i	$0.193789\pi$
$152$	0	0
$153$	3.14683e7	0.710318
$154$	0	0
$155$	−2.42485e6	−0.0523027
$156$	0	0
$157$	6.27066e6	0.129320	0.0646598	−	0.997907i	$-0.479404\pi$
$157$	6.27066e6	0.129320	0.0646598	+	0.997907i	$0.479404\pi$
$158$	0	0
$159$	8.19599e6	0.161701
$160$	0	0
$161$	8.55551e7	1.61568
$162$	0	0
$163$	−7.98890e7	−1.44488	−0.722438	−	0.691436i	$-0.756978\pi$
$163$	−7.98890e7	−1.44488	−0.722438	+	0.691436i	$0.756978\pi$
$164$	0	0
$165$	−1.23453e7	−0.213948
$166$	0	0
$167$	2.27801e7	0.378484	0.189242	−	0.981930i	$-0.439397\pi$
$167$	2.27801e7	0.378484	0.189242	+	0.981930i	$0.439397\pi$
$168$	0	0
$169$	5.89437e7	0.939364
$170$	0	0
$171$	6.73614e7	1.03021
$172$	0	0
$173$	1.69218e7	0.248477	0.124238	−	0.992252i	$-0.460351\pi$
$173$	1.69218e7	0.248477	0.124238	+	0.992252i	$0.460351\pi$
$174$	0	0
$175$	2.59586e7	0.366141
$176$	0	0
$177$	−8.74921e6	−0.118594
$178$	0	0
$179$	−2.62419e6	−0.0341987	−0.0170994	−	0.999854i	$-0.505443\pi$
$179$	−2.62419e6	−0.0341987	−0.0170994	+	0.999854i	$0.505443\pi$
$180$	0	0
$181$	−5.95058e7	−0.745906	−0.372953	−	0.927850i	$-0.621655\pi$
$181$	−5.95058e7	−0.745906	−0.372953	+	0.927850i	$0.621655\pi$
$182$	0	0
$183$	3.97454e7	0.479411
$184$	0	0
$185$	−7.21044e7	−0.837260
$186$	0	0
$187$	9.41927e7	1.05335
$188$	0	0
$189$	−1.18635e8	−1.27819
$190$	0	0
$191$	3.23352e7	0.335783	0.167891	−	0.985806i	$-0.446304\pi$
$191$	3.23352e7	0.335783	0.167891	+	0.985806i	$0.446304\pi$
$192$	0	0
$193$	6.24103e6	0.0624893	0.0312446	−	0.999512i	$-0.490053\pi$
$193$	6.24103e6	0.0624893	0.0312446	+	0.999512i	$0.490053\pi$
$194$	0	0
$195$	−2.42204e7	−0.233916
$196$	0	0
$197$	−1.63779e8	−1.52625	−0.763125	−	0.646251i	$-0.776336\pi$
$197$	−1.63779e8	−1.52625	−0.763125	+	0.646251i	$0.776336\pi$
$198$	0	0
$199$	−6.10438e7	−0.549106	−0.274553	−	0.961572i	$-0.588530\pi$
$199$	−6.10438e7	−0.549106	−0.274553	+	0.961572i	$0.588530\pi$
$200$	0	0
$201$	−3.39215e7	−0.294638
$202$	0	0
$203$	3.27746e8	2.74980
$204$	0	0
$205$	−6.71447e7	−0.544344
$206$	0	0
$207$	−9.67369e7	−0.758047
$208$	0	0
$209$	2.01630e8	1.52772
$210$	0	0
$211$	1.38165e7	0.101253	0.0506267	−	0.998718i	$-0.483878\pi$
$211$	1.38165e7	0.101253	0.0506267	+	0.998718i	$0.483878\pi$
$212$	0	0
$213$	4.23901e7	0.300563
$214$	0	0
$215$	−727475.	−0.00499210
$216$	0	0
$217$	−3.22282e7	−0.214105
$218$	0	0
$219$	4.62940e7	0.297831
$220$	0	0
$221$	1.84798e8	1.15166
$222$	0	0
$223$	4.37107e7	0.263949	0.131975	−	0.991253i	$-0.457868\pi$
$223$	4.37107e7	0.263949	0.131975	+	0.991253i	$0.457868\pi$
$224$	0	0
$225$	−2.93513e7	−0.171786
$226$	0	0
$227$	−3.83906e7	−0.217838	−0.108919	−	0.994051i	$-0.534739\pi$
$227$	−3.83906e7	−0.217838	−0.108919	+	0.994051i	$0.534739\pi$
$228$	0	0
$229$	2.18239e8	1.20090	0.600452	−	0.799661i	$-0.294987\pi$
$229$	2.18239e8	1.20090	0.600452	+	0.799661i	$0.294987\pi$
$230$	0	0
$231$	−1.64079e8	−0.875811
$232$	0	0
$233$	2.52167e8	1.30600	0.653000	−	0.757358i	$-0.273510\pi$
$233$	2.52167e8	1.30600	0.653000	+	0.757358i	$0.273510\pi$
$234$	0	0
$235$	−1.30332e8	−0.655109
$236$	0	0
$237$	−1.47769e8	−0.721049
$238$	0	0
$239$	4.04391e7	0.191606	0.0958030	−	0.995400i	$-0.469458\pi$
$239$	4.04391e7	0.191606	0.0958030	+	0.995400i	$0.469458\pi$
$240$	0	0
$241$	5.56292e6	0.0256002	0.0128001	−	0.999918i	$-0.495925\pi$
$241$	5.56292e6	0.0256002	0.0128001	+	0.999918i	$0.495925\pi$
$242$	0	0
$243$	2.06300e8	0.922311
$244$	0	0
$245$	2.42067e8	1.05161
$246$	0	0
$247$	3.95581e8	1.67030
$248$	0	0
$249$	−9.69526e7	−0.397981
$250$	0	0
$251$	−2.65218e8	−1.05863	−0.529316	−	0.848425i	$-0.677551\pi$
$251$	−2.65218e8	−1.05863	−0.529316	+	0.848425i	$0.677551\pi$
$252$	0	0
$253$	−2.89559e8	−1.12413
$254$	0	0
$255$	−3.67803e7	−0.138907
$256$	0	0
$257$	−2.09163e7	−0.0768634	−0.0384317	−	0.999261i	$-0.512236\pi$
$257$	−2.09163e7	−0.0768634	−0.0384317	+	0.999261i	$0.512236\pi$
$258$	0	0
$259$	−9.58325e8	−3.42739
$260$	0	0
$261$	−3.70581e8	−1.29016
$262$	0	0
$263$	−4.64938e8	−1.57598	−0.787988	−	0.615691i	$-0.788877\pi$
$263$	−4.64938e8	−1.57598	−0.787988	+	0.615691i	$0.788877\pi$
$264$	0	0
$265$	5.83273e7	0.192536
$266$	0	0
$267$	−1.04885e8	−0.337228
$268$	0	0
$269$	−2.21015e8	−0.692291	−0.346145	−	0.938181i	$-0.612510\pi$
$269$	−2.21015e8	−0.692291	−0.346145	+	0.938181i	$0.612510\pi$
$270$	0	0
$271$	4.87299e8	1.48731	0.743657	−	0.668561i	$-0.233089\pi$
$271$	4.87299e8	1.48731	0.743657	+	0.668561i	$0.233089\pi$
$272$	0	0
$273$	−3.21908e8	−0.957553
$274$	0	0
$275$	−8.78560e7	−0.254746
$276$	0	0
$277$	5.05955e8	1.43032	0.715160	−	0.698961i	$-0.246354\pi$
$277$	5.05955e8	1.43032	0.715160	+	0.698961i	$0.246354\pi$
$278$	0	0
$279$	3.64403e7	0.100454
$280$	0	0
$281$	−1.13423e8	−0.304950	−0.152475	−	0.988307i	$-0.548724\pi$
$281$	−1.13423e8	−0.304950	−0.152475	+	0.988307i	$0.548724\pi$
$282$	0	0
$283$	1.04096e8	0.273012	0.136506	−	0.990639i	$-0.456413\pi$
$283$	1.04096e8	0.273012	0.136506	+	0.990639i	$0.456413\pi$
$284$	0	0
$285$	−7.87324e7	−0.201464
$286$	0	0
$287$	−8.92406e8	−2.22831
$288$	0	0
$289$	−1.29710e8	−0.316105
$290$	0	0
$291$	2.88867e8	0.687184
$292$	0	0
$293$	4.54809e8	1.05631	0.528156	−	0.849147i	$-0.322883\pi$
$293$	4.54809e8	1.05631	0.528156	+	0.849147i	$0.322883\pi$
$294$	0	0
$295$	−6.22643e7	−0.141209
$296$	0	0
$297$	4.01517e8	0.889315
$298$	0	0
$299$	−5.68089e8	−1.22904
$300$	0	0
$301$	−9.66872e6	−0.0204355
$302$	0	0
$303$	−1.21214e8	−0.250324
$304$	0	0
$305$	2.82850e8	0.570831
$306$	0	0
$307$	3.95297e8	0.779721	0.389861	−	0.920874i	$-0.372523\pi$
$307$	3.95297e8	0.779721	0.389861	+	0.920874i	$0.372523\pi$
$308$	0	0
$309$	1.43242e8	0.276194
$310$	0	0
$311$	−6.41605e7	−0.120950	−0.0604751	−	0.998170i	$-0.519262\pi$
$311$	−6.41605e7	−0.120950	−0.0604751	+	0.998170i	$0.519262\pi$
$312$	0	0
$313$	−4.16252e8	−0.767274	−0.383637	−	0.923484i	$-0.625329\pi$
$313$	−4.16252e8	−0.767274	−0.383637	+	0.923484i	$0.625329\pi$
$314$	0	0
$315$	−3.90102e8	−0.703220
$316$	0	0
$317$	2.06280e8	0.363706	0.181853	−	0.983326i	$-0.441790\pi$
$317$	2.06280e8	0.363706	0.181853	+	0.983326i	$0.441790\pi$
$318$	0	0
$319$	−1.10925e9	−1.91320
$320$	0	0
$321$	−2.09117e8	−0.352876
$322$	0	0
$323$	6.00716e8	0.991883
$324$	0	0
$325$	−1.72366e8	−0.278522
$326$	0	0
$327$	−4.64504e7	−0.0734636
$328$	0	0
$329$	−1.73221e9	−2.68174
$330$	0	0
$331$	−2.51051e8	−0.380508	−0.190254	−	0.981735i	$-0.560931\pi$
$331$	−2.51051e8	−0.380508	−0.190254	+	0.981735i	$0.560931\pi$
$332$	0	0
$333$	1.08357e9	1.60807
$334$	0	0
$335$	−2.41404e8	−0.350823
$336$	0	0
$337$	1.85482e8	0.263996	0.131998	−	0.991250i	$-0.457861\pi$
$337$	1.85482e8	0.263996	0.131998	+	0.991250i	$0.457861\pi$
$338$	0	0
$339$	−2.62939e8	−0.366569
$340$	0	0
$341$	1.09075e8	0.148966
$342$	0	0
$343$	1.84908e9	2.47415
$344$	0	0
$345$	1.13067e8	0.148241
$346$	0	0
$347$	−2.85208e8	−0.366445	−0.183223	−	0.983071i	$-0.558653\pi$
$347$	−2.85208e8	−0.366445	−0.183223	+	0.983071i	$0.558653\pi$
$348$	0	0
$349$	−1.67666e8	−0.211133	−0.105567	−	0.994412i	$-0.533666\pi$
$349$	−1.67666e8	−0.211133	−0.105567	+	0.994412i	$0.533666\pi$
$350$	0	0
$351$	7.87741e8	0.972318
$352$	0	0
$353$	−4.51601e8	−0.546441	−0.273220	−	0.961951i	$-0.588089\pi$
$353$	−4.51601e8	−0.546441	−0.273220	+	0.961951i	$0.588089\pi$
$354$	0	0
$355$	3.01672e8	0.357878
$356$	0	0
$357$	−4.88840e8	−0.568627
$358$	0	0
$359$	−1.55976e9	−1.77921	−0.889603	−	0.456735i	$-0.849019\pi$
$359$	−1.55976e9	−1.77921	−0.889603	+	0.456735i	$0.849019\pi$
$360$	0	0
$361$	3.92029e8	0.438574
$362$	0	0
$363$	2.13034e8	0.233763
$364$	0	0
$365$	3.29454e8	0.354625
$366$	0	0
$367$	−1.98093e8	−0.209189	−0.104594	−	0.994515i	$-0.533354\pi$
$367$	−1.98093e8	−0.209189	−0.104594	+	0.994515i	$0.533354\pi$
$368$	0	0
$369$	1.00904e9	1.04548
$370$	0	0
$371$	7.75216e8	0.788159
$372$	0	0
$373$	8.64758e8	0.862807	0.431404	−	0.902159i	$-0.358018\pi$
$373$	8.64758e8	0.862807	0.431404	+	0.902159i	$0.358018\pi$
$374$	0	0
$375$	3.43060e7	0.0335939
$376$	0	0
$377$	−2.17625e9	−2.09177
$378$	0	0
$379$	8.26044e8	0.779410	0.389705	−	0.920940i	$-0.372577\pi$
$379$	8.26044e8	0.779410	0.389705	+	0.920940i	$0.372577\pi$
$380$	0	0
$381$	−1.79470e8	−0.166247
$382$	0	0
$383$	1.79537e9	1.63289	0.816446	−	0.577421i	$-0.195941\pi$
$383$	1.79537e9	1.63289	0.816446	+	0.577421i	$0.195941\pi$
$384$	0	0
$385$	−1.16768e9	−1.04282
$386$	0	0
$387$	1.09324e7	0.00958797
$388$	0	0
$389$	1.89068e9	1.62852	0.814262	−	0.580498i	$-0.197142\pi$
$389$	1.89068e9	1.62852	0.814262	+	0.580498i	$0.197142\pi$
$390$	0	0
$391$	−8.62682e8	−0.729848
$392$	0	0
$393$	2.09580e8	0.174171
$394$	0	0
$395$	−1.05161e9	−0.858547
$396$	0	0
$397$	4.47209e8	0.358710	0.179355	−	0.983784i	$-0.442599\pi$
$397$	4.47209e8	0.358710	0.179355	+	0.983784i	$0.442599\pi$
$398$	0	0
$399$	−1.04642e9	−0.824706
$400$	0	0
$401$	5.02001e8	0.388776	0.194388	−	0.980925i	$-0.437728\pi$
$401$	5.02001e8	0.388776	0.194388	+	0.980925i	$0.437728\pi$
$402$	0	0
$403$	2.13996e8	0.162869
$404$	0	0
$405$	3.56746e8	0.266850
$406$	0	0
$407$	3.24342e9	2.38464
$408$	0	0
$409$	1.85997e9	1.34423	0.672115	−	0.740446i	$-0.265386\pi$
$409$	1.85997e9	1.34423	0.672115	+	0.740446i	$0.265386\pi$
$410$	0	0
$411$	3.78229e8	0.268726
$412$	0	0
$413$	−8.27542e8	−0.578049
$414$	0	0
$415$	−6.89969e8	−0.473872
$416$	0	0
$417$	3.38735e8	0.228762
$418$	0	0
$419$	−1.24339e9	−0.825770	−0.412885	−	0.910783i	$-0.635479\pi$
$419$	−1.24339e9	−0.825770	−0.412885	+	0.910783i	$0.635479\pi$
$420$	0	0
$421$	1.46595e9	0.957487	0.478744	−	0.877955i	$-0.341092\pi$
$421$	1.46595e9	0.957487	0.478744	+	0.877955i	$0.341092\pi$
$422$	0	0
$423$	1.95861e9	1.25822
$424$	0	0
$425$	−2.61749e8	−0.165396
$426$	0	0
$427$	3.75931e9	2.33674
$428$	0	0
$429$	1.08949e9	0.666227
$430$	0	0
$431$	3.23266e9	1.94487	0.972434	−	0.233179i	$-0.0749129\pi$
$431$	3.23266e9	1.94487	0.972434	+	0.233179i	$0.0749129\pi$
$432$	0	0
$433$	−2.13634e9	−1.26463	−0.632313	−	0.774713i	$-0.717894\pi$
$433$	−2.13634e9	−1.26463	−0.632313	+	0.774713i	$0.717894\pi$
$434$	0	0
$435$	4.33138e8	0.252298
$436$	0	0
$437$	−1.84667e9	−1.05853
$438$	0	0
$439$	−1.75840e9	−0.991957	−0.495978	−	0.868335i	$-0.665190\pi$
$439$	−1.75840e9	−0.991957	−0.495978	+	0.868335i	$0.665190\pi$
$440$	0	0
$441$	−3.63776e9	−2.01975
$442$	0	0
$443$	1.03894e9	0.567774	0.283887	−	0.958858i	$-0.408376\pi$
$443$	1.03894e9	0.567774	0.283887	+	0.958858i	$0.408376\pi$
$444$	0	0
$445$	−7.46419e8	−0.401534
$446$	0	0
$447$	−5.21190e8	−0.276007
$448$	0	0
$449$	−3.02217e7	−0.0157564	−0.00787821	−	0.999969i	$-0.502508\pi$
$449$	−3.02217e7	−0.0157564	−0.00787821	+	0.999969i	$0.502508\pi$
$450$	0	0
$451$	3.02032e9	1.55037
$452$	0	0
$453$	1.21921e9	0.616218
$454$	0	0
$455$	−2.29088e9	−1.14015
$456$	0	0
$457$	4.26007e8	0.208790	0.104395	−	0.994536i	$-0.466709\pi$
$457$	4.26007e8	0.208790	0.104395	+	0.994536i	$0.466709\pi$
$458$	0	0
$459$	1.19624e9	0.577395
$460$	0	0
$461$	−1.78754e9	−0.849772	−0.424886	−	0.905247i	$-0.639686\pi$
$461$	−1.78754e9	−0.849772	−0.424886	+	0.905247i	$0.639686\pi$
$462$	0	0
$463$	2.52187e9	1.18084	0.590419	−	0.807097i	$-0.298963\pi$
$463$	2.52187e9	1.18084	0.590419	+	0.807097i	$0.298963\pi$
$464$	0	0
$465$	−4.25917e7	−0.0196444
$466$	0	0
$467$	−3.60291e9	−1.63699	−0.818493	−	0.574517i	$-0.805190\pi$
$467$	−3.60291e9	−1.63699	−0.818493	+	0.574517i	$0.805190\pi$
$468$	0	0
$469$	−3.20846e9	−1.43612
$470$	0	0
$471$	1.10142e8	0.0485713
$472$	0	0
$473$	3.27235e7	0.0142182
$474$	0	0
$475$	−5.60304e8	−0.239881
$476$	0	0
$477$	−8.76534e8	−0.369789
$478$	0	0
$479$	−2.87149e9	−1.19381	−0.596903	−	0.802314i	$-0.703602\pi$
$479$	−2.87149e9	−1.19381	−0.596903	+	0.802314i	$0.703602\pi$
$480$	0	0
$481$	6.36331e9	2.60721
$482$	0	0
$483$	1.50275e9	0.606835
$484$	0	0
$485$	2.05574e9	0.818224
$486$	0	0
$487$	−2.48078e9	−0.973277	−0.486638	−	0.873604i	$-0.661777\pi$
$487$	−2.48078e9	−0.973277	−0.486638	+	0.873604i	$0.661777\pi$
$488$	0	0
$489$	−1.40322e9	−0.542683
$490$	0	0
$491$	−3.61256e9	−1.37731	−0.688653	−	0.725091i	$-0.741797\pi$
$491$	−3.61256e9	−1.37731	−0.688653	+	0.725091i	$0.741797\pi$
$492$	0	0
$493$	−3.30478e9	−1.24216
$494$	0	0
$495$	1.32029e9	0.489272
$496$	0	0
$497$	4.00946e9	1.46500
$498$	0	0
$499$	−3.78849e9	−1.36494	−0.682471	−	0.730912i	$-0.739095\pi$
$499$	−3.78849e9	−1.36494	−0.682471	+	0.730912i	$0.739095\pi$
$500$	0	0
$501$	4.00125e8	0.142155
$502$	0	0
$503$	−8.09636e8	−0.283663	−0.141831	−	0.989891i	$-0.545299\pi$
$503$	−8.09636e8	−0.283663	−0.141831	+	0.989891i	$0.545299\pi$
$504$	0	0
$505$	−8.62625e8	−0.298059
$506$	0	0
$507$	1.03533e9	0.352817
$508$	0	0
$509$	−2.74098e9	−0.921284	−0.460642	−	0.887586i	$-0.652381\pi$
$509$	−2.74098e9	−0.921284	−0.460642	+	0.887586i	$0.652381\pi$
$510$	0	0
$511$	4.37870e9	1.45169
$512$	0	0
$513$	2.56068e9	0.837423
$514$	0	0
$515$	1.01939e9	0.328863
$516$	0	0
$517$	5.86263e9	1.86584
$518$	0	0
$519$	2.97226e8	0.0933257
$520$	0	0
$521$	3.08219e9	0.954834	0.477417	−	0.878677i	$-0.341573\pi$
$521$	3.08219e9	0.954834	0.477417	+	0.878677i	$0.341573\pi$
$522$	0	0
$523$	4.63997e9	1.41827	0.709135	−	0.705072i	$-0.249085\pi$
$523$	4.63997e9	1.41827	0.709135	+	0.705072i	$0.249085\pi$
$524$	0	0
$525$	4.55954e8	0.137519
$526$	0	0
$527$	3.24968e8	0.0967173
$528$	0	0
$529$	−7.52847e8	−0.221112
$530$	0	0
$531$	9.35699e8	0.271210
$532$	0	0
$533$	5.92561e9	1.69507
$534$	0	0
$535$	−1.48819e9	−0.420167
$536$	0	0
$537$	−4.60931e7	−0.0128447
$538$	0	0
$539$	−1.08887e10	−2.99514
$540$	0	0
$541$	−4.21606e9	−1.14477	−0.572383	−	0.819987i	$-0.693981\pi$
$541$	−4.21606e9	−1.14477	−0.572383	+	0.819987i	$0.693981\pi$
$542$	0	0
$543$	−1.04520e9	−0.280156
$544$	0	0
$545$	−3.30567e8	−0.0874725
$546$	0	0
$547$	−3.74991e9	−0.979636	−0.489818	−	0.871825i	$-0.662937\pi$
$547$	−3.74991e9	−0.979636	−0.489818	+	0.871825i	$0.662937\pi$
$548$	0	0
$549$	−4.25064e9	−1.09635
$550$	0	0
$551$	−7.07425e9	−1.80156
$552$	0	0
$553$	−1.39767e10	−3.51453
$554$	0	0
$555$	−1.26649e9	−0.314468
$556$	0	0
$557$	−1.55904e9	−0.382265	−0.191132	−	0.981564i	$-0.561216\pi$
$557$	−1.55904e9	−0.382265	−0.191132	+	0.981564i	$0.561216\pi$
$558$	0	0
$559$	6.42006e7	0.0155453
$560$	0	0
$561$	1.65446e9	0.395628
$562$	0	0
$563$	−8.01780e8	−0.189355	−0.0946773	−	0.995508i	$-0.530182\pi$
$563$	−8.01780e8	−0.189355	−0.0946773	+	0.995508i	$0.530182\pi$
$564$	0	0
$565$	−1.87122e9	−0.436471
$566$	0	0
$567$	4.74144e9	1.09237
$568$	0	0
$569$	1.65652e9	0.376967	0.188484	−	0.982076i	$-0.439643\pi$
$569$	1.65652e9	0.376967	0.188484	+	0.982076i	$0.439643\pi$
$570$	0	0
$571$	−8.44462e8	−0.189825	−0.0949126	−	0.995486i	$-0.530257\pi$
$571$	−8.44462e8	−0.189825	−0.0949126	+	0.995486i	$0.530257\pi$
$572$	0	0
$573$	5.67956e8	0.126117
$574$	0	0
$575$	8.04646e8	0.176509
$576$	0	0
$577$	5.72151e9	1.23992	0.619962	−	0.784631i	$-0.287148\pi$
$577$	5.72151e9	1.23992	0.619962	+	0.784631i	$0.287148\pi$
$578$	0	0
$579$	1.09622e8	0.0234704
$580$	0	0
$581$	−9.17024e9	−1.93983
$582$	0	0
$583$	−2.62369e9	−0.548369
$584$	0	0
$585$	2.59029e9	0.534937
$586$	0	0
$587$	−3.55364e9	−0.725171	−0.362585	−	0.931951i	$-0.618106\pi$
$587$	−3.55364e9	−0.725171	−0.362585	+	0.931951i	$0.618106\pi$
$588$	0	0
$589$	6.95631e8	0.140273
$590$	0	0
$591$	−2.87672e9	−0.573246
$592$	0	0
$593$	7.22540e9	1.42289	0.711443	−	0.702744i	$-0.248042\pi$
$593$	7.22540e9	1.42289	0.711443	+	0.702744i	$0.248042\pi$
$594$	0	0
$595$	−3.47886e9	−0.677060
$596$	0	0
$597$	−1.07221e9	−0.206239
$598$	0	0
$599$	6.11590e7	0.0116270	0.00581348	−	0.999983i	$-0.498150\pi$
$599$	6.11590e7	0.0116270	0.00581348	+	0.999983i	$0.498150\pi$
$600$	0	0
$601$	6.55031e9	1.23084	0.615419	−	0.788200i	$-0.288987\pi$
$601$	6.55031e9	1.23084	0.615419	+	0.788200i	$0.288987\pi$
$602$	0	0
$603$	3.62779e9	0.673801
$604$	0	0
$605$	1.51607e9	0.278340
$606$	0	0
$607$	−4.94498e9	−0.897436	−0.448718	−	0.893673i	$-0.648119\pi$
$607$	−4.94498e9	−0.897436	−0.448718	+	0.893673i	$0.648119\pi$
$608$	0	0
$609$	5.75675e9	1.03280
$610$	0	0
$611$	1.15020e10	2.03999
$612$	0	0
$613$	8.80012e8	0.154304	0.0771520	−	0.997019i	$-0.475417\pi$
$613$	8.80012e8	0.154304	0.0771520	+	0.997019i	$0.475417\pi$
$614$	0	0
$615$	−1.17937e9	−0.204451
$616$	0	0
$617$	−6.76308e8	−0.115917	−0.0579584	−	0.998319i	$-0.518459\pi$
$617$	−6.76308e8	−0.115917	−0.0579584	+	0.998319i	$0.518459\pi$
$618$	0	0
$619$	−2.44000e9	−0.413497	−0.206748	−	0.978394i	$-0.566288\pi$
$619$	−2.44000e9	−0.413497	−0.206748	+	0.978394i	$0.566288\pi$
$620$	0	0
$621$	−3.67737e9	−0.616192
$622$	0	0
$623$	−9.92051e9	−1.64371
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	3.54156e9	0.573797
$628$	0	0
$629$	9.66312e9	1.54825
$630$	0	0
$631$	−2.83883e9	−0.449817	−0.224909	−	0.974380i	$-0.572208\pi$
$631$	−2.83883e9	−0.449817	−0.224909	+	0.974380i	$0.572208\pi$
$632$	0	0
$633$	2.42682e8	0.0380299
$634$	0	0
$635$	−1.27721e9	−0.197949
$636$	0	0
$637$	−2.13628e10	−3.27469
$638$	0	0
$639$	−4.53348e9	−0.687351
$640$	0	0
$641$	9.22881e8	0.138402	0.0692011	−	0.997603i	$-0.477955\pi$
$641$	9.22881e8	0.138402	0.0692011	+	0.997603i	$0.477955\pi$
$642$	0	0
$643$	1.28831e9	0.191109	0.0955544	−	0.995424i	$-0.469538\pi$
$643$	1.28831e9	0.191109	0.0955544	+	0.995424i	$0.469538\pi$
$644$	0	0
$645$	−1.27778e7	−0.00187499
$646$	0	0
$647$	1.73640e9	0.252050	0.126025	−	0.992027i	$-0.459778\pi$
$647$	1.73640e9	0.252050	0.126025	+	0.992027i	$0.459778\pi$
$648$	0	0
$649$	2.80079e9	0.402183
$650$	0	0
$651$	−5.66078e8	−0.0804160
$652$	0	0
$653$	−4.96284e8	−0.0697484	−0.0348742	−	0.999392i	$-0.511103\pi$
$653$	−4.96284e8	−0.0697484	−0.0348742	+	0.999392i	$0.511103\pi$
$654$	0	0
$655$	1.49149e9	0.207385
$656$	0	0
$657$	−4.95099e9	−0.681103
$658$	0	0
$659$	3.10480e9	0.422606	0.211303	−	0.977421i	$-0.432229\pi$
$659$	3.10480e9	0.422606	0.211303	+	0.977421i	$0.432229\pi$
$660$	0	0
$661$	4.04807e9	0.545183	0.272592	−	0.962130i	$-0.412119\pi$
$661$	4.04807e9	0.545183	0.272592	+	0.962130i	$0.412119\pi$
$662$	0	0
$663$	3.24591e9	0.432553
$664$	0	0
$665$	−7.44689e9	−0.981972
$666$	0	0
$667$	1.01592e10	1.32563
$668$	0	0
$669$	7.67763e8	0.0991370
$670$	0	0
$671$	−1.27233e10	−1.62581
$672$	0	0
$673$	−6.38243e9	−0.807111	−0.403556	−	0.914955i	$-0.632226\pi$
$673$	−6.38243e9	−0.807111	−0.403556	+	0.914955i	$0.632226\pi$
$674$	0	0
$675$	−1.11576e9	−0.139640
$676$	0	0
$677$	−7.76270e9	−0.961507	−0.480753	−	0.876856i	$-0.659637\pi$
$677$	−7.76270e9	−0.961507	−0.480753	+	0.876856i	$0.659637\pi$
$678$	0	0
$679$	2.73224e10	3.34946
$680$	0	0
$681$	−6.74318e8	−0.0818182
$682$	0	0
$683$	−1.01635e10	−1.22060	−0.610298	−	0.792172i	$-0.708950\pi$
$683$	−1.01635e10	−1.22060	−0.610298	+	0.792172i	$0.708950\pi$
$684$	0	0
$685$	2.69169e9	0.319969
$686$	0	0
$687$	3.83330e9	0.451049
$688$	0	0
$689$	−5.14746e9	−0.599550
$690$	0	0
$691$	−4.41960e9	−0.509577	−0.254788	−	0.966997i	$-0.582006\pi$
$691$	−4.41960e9	−0.509577	−0.254788	+	0.966997i	$0.582006\pi$
$692$	0	0
$693$	1.75477e10	2.00287
$694$	0	0
$695$	2.41063e9	0.272385
$696$	0	0
$697$	8.99845e9	1.00659
$698$	0	0
$699$	4.42924e9	0.490522
$700$	0	0
$701$	−1.19204e10	−1.30701	−0.653505	−	0.756922i	$-0.726702\pi$
$701$	−1.19204e10	−1.30701	−0.653505	+	0.756922i	$0.726702\pi$
$702$	0	0
$703$	2.06850e10	2.24549
$704$	0	0
$705$	−2.28924e9	−0.246053
$706$	0	0
$707$	−1.14650e10	−1.22013
$708$	0	0
$709$	3.25551e9	0.343050	0.171525	−	0.985180i	$-0.445131\pi$
$709$	3.25551e9	0.343050	0.171525	+	0.985180i	$0.445131\pi$
$710$	0	0
$711$	1.58034e10	1.64895
$712$	0	0
$713$	−9.98988e8	−0.103216
$714$	0	0
$715$	7.75341e9	0.793271
$716$	0	0
$717$	7.10299e8	0.0719655
$718$	0	0
$719$	−1.11076e10	−1.11447	−0.557233	−	0.830356i	$-0.688137\pi$
$719$	−1.11076e10	−1.11447	−0.557233	+	0.830356i	$0.688137\pi$
$720$	0	0
$721$	1.35485e10	1.34622
$722$	0	0
$723$	9.77107e7	0.00961520
$724$	0	0
$725$	3.08245e9	0.300409
$726$	0	0
$727$	−5.99593e9	−0.578744	−0.289372	−	0.957217i	$-0.593446\pi$
$727$	−5.99593e9	−0.578744	−0.289372	+	0.957217i	$0.593446\pi$
$728$	0	0
$729$	−2.61804e9	−0.250282
$730$	0	0
$731$	9.74931e7	0.00923130
$732$	0	0
$733$	1.60252e10	1.50293	0.751467	−	0.659771i	$-0.229347\pi$
$733$	1.60252e10	1.50293	0.751467	+	0.659771i	$0.229347\pi$
$734$	0	0
$735$	4.25183e9	0.394976
$736$	0	0
$737$	1.08589e10	0.999195
$738$	0	0
$739$	4.63052e9	0.422060	0.211030	−	0.977480i	$-0.432318\pi$
$739$	4.63052e9	0.422060	0.211030	+	0.977480i	$0.432318\pi$
$740$	0	0
$741$	6.94824e9	0.627352
$742$	0	0
$743$	3.84481e9	0.343886	0.171943	−	0.985107i	$-0.444996\pi$
$743$	3.84481e9	0.343886	0.171943	+	0.985107i	$0.444996\pi$
$744$	0	0
$745$	−3.70908e9	−0.328639
$746$	0	0
$747$	1.03688e10	0.910133
$748$	0	0
$749$	−1.97793e10	−1.71998
$750$	0	0
$751$	1.90095e10	1.63769	0.818843	−	0.574017i	$-0.194616\pi$
$751$	1.90095e10	1.63769	0.818843	+	0.574017i	$0.194616\pi$
$752$	0	0
$753$	−4.65847e9	−0.397613
$754$	0	0
$755$	8.67657e9	0.733726
$756$	0	0
$757$	1.70131e10	1.42544	0.712720	−	0.701449i	$-0.247463\pi$
$757$	1.70131e10	1.42544	0.712720	+	0.701449i	$0.247463\pi$
$758$	0	0
$759$	−5.08600e9	−0.422212
$760$	0	0
$761$	−1.88582e10	−1.55115	−0.775575	−	0.631255i	$-0.782540\pi$
$761$	−1.88582e10	−1.55115	−0.775575	+	0.631255i	$0.782540\pi$
$762$	0	0
$763$	−4.39350e9	−0.358075
$764$	0	0
$765$	3.93353e9	0.317664
$766$	0	0
$767$	5.49491e9	0.439720
$768$	0	0
$769$	1.49558e10	1.18595	0.592976	−	0.805220i	$-0.297953\pi$
$769$	1.49558e10	1.18595	0.592976	+	0.805220i	$0.297953\pi$
$770$	0	0
$771$	−3.67388e8	−0.0288692
$772$	0	0
$773$	9.47016e9	0.737444	0.368722	−	0.929540i	$-0.379795\pi$
$773$	9.47016e9	0.737444	0.368722	+	0.929540i	$0.379795\pi$
$774$	0	0
$775$	−3.03107e8	−0.0233905
$776$	0	0
$777$	−1.68326e10	−1.28730
$778$	0	0
$779$	1.92622e10	1.45990
$780$	0	0
$781$	−1.35699e10	−1.01929
$782$	0	0
$783$	−1.40873e10	−1.04873
$784$	0	0
$785$	7.83832e8	0.0578335
$786$	0	0
$787$	2.34996e9	0.171849	0.0859247	−	0.996302i	$-0.472616\pi$
$787$	2.34996e9	0.171849	0.0859247	+	0.996302i	$0.472616\pi$
$788$	0	0
$789$	−8.16647e9	−0.591923
$790$	0	0
$791$	−2.48700e10	−1.78673
$792$	0	0
$793$	−2.49619e10	−1.77755
$794$	0	0
$795$	1.02450e9	0.0723147
$796$	0	0
$797$	1.37072e10	0.959057	0.479529	−	0.877526i	$-0.340808\pi$
$797$	1.37072e10	0.959057	0.479529	+	0.877526i	$0.340808\pi$
$798$	0	0
$799$	1.74665e10	1.21141
$800$	0	0
$801$	1.12171e10	0.771199
$802$	0	0
$803$	−1.48196e10	−1.01002
$804$	0	0
$805$	1.06944e10	0.722554
$806$	0	0
$807$	−3.88205e9	−0.260018
$808$	0	0
$809$	−2.45778e10	−1.63201	−0.816006	−	0.578043i	$-0.803817\pi$
$809$	−2.45778e10	−1.63201	−0.816006	+	0.578043i	$0.803817\pi$
$810$	0	0
$811$	4.88640e9	0.321674	0.160837	−	0.986981i	$-0.448581\pi$
$811$	4.88640e9	0.321674	0.160837	+	0.986981i	$0.448581\pi$
$812$	0	0
$813$	8.55924e9	0.558622
$814$	0	0
$815$	−9.98613e9	−0.646168
$816$	0	0
$817$	2.08695e8	0.0133886
$818$	0	0
$819$	3.44270e10	2.18981
$820$	0	0
$821$	−3.28588e9	−0.207229	−0.103615	−	0.994618i	$-0.533041\pi$
$821$	−3.28588e9	−0.207229	−0.103615	+	0.994618i	$0.533041\pi$
$822$	0	0
$823$	−2.64607e10	−1.65463	−0.827317	−	0.561735i	$-0.810134\pi$
$823$	−2.64607e10	−1.65463	−0.827317	+	0.561735i	$0.810134\pi$
$824$	0	0
$825$	−1.54316e9	−0.0956803
$826$	0	0
$827$	−2.81695e10	−1.73185	−0.865925	−	0.500174i	$-0.833269\pi$
$827$	−2.81695e10	−1.73185	−0.865925	+	0.500174i	$0.833269\pi$
$828$	0	0
$829$	−1.06563e10	−0.649629	−0.324814	−	0.945778i	$-0.605302\pi$
$829$	−1.06563e10	−0.649629	−0.324814	+	0.945778i	$0.605302\pi$
$830$	0	0
$831$	8.88694e9	0.537215
$832$	0	0
$833$	−3.24408e10	−1.94462
$834$	0	0
$835$	2.84751e9	0.169263
$836$	0	0
$837$	1.38525e9	0.0816560
$838$	0	0
$839$	−1.41352e9	−0.0826295	−0.0413147	−	0.999146i	$-0.513155\pi$
$839$	−1.41352e9	−0.0826295	−0.0413147	+	0.999146i	$0.513155\pi$
$840$	0	0
$841$	2.16683e10	1.25614
$842$	0	0
$843$	−1.99223e9	−0.114536
$844$	0	0
$845$	7.36796e9	0.420096
$846$	0	0
$847$	2.01498e10	1.13940
$848$	0	0
$849$	1.82841e9	0.102541
$850$	0	0
$851$	−2.97055e10	−1.65228
$852$	0	0
$853$	−1.12037e10	−0.618074	−0.309037	−	0.951050i	$-0.600007\pi$
$853$	−1.12037e10	−0.618074	−0.309037	+	0.951050i	$0.600007\pi$
$854$	0	0
$855$	8.42017e9	0.460723
$856$	0	0
$857$	2.27859e10	1.23661	0.618306	−	0.785937i	$-0.287819\pi$
$857$	2.27859e10	1.23661	0.618306	+	0.785937i	$0.287819\pi$
$858$	0	0
$859$	1.80638e10	0.972374	0.486187	−	0.873855i	$-0.338387\pi$
$859$	1.80638e10	0.972374	0.486187	+	0.873855i	$0.338387\pi$
$860$	0	0
$861$	−1.56748e10	−0.836934
$862$	0	0
$863$	−8.77045e9	−0.464498	−0.232249	−	0.972656i	$-0.574608\pi$
$863$	−8.77045e9	−0.464498	−0.232249	+	0.972656i	$0.574608\pi$
$864$	0	0
$865$	2.11523e9	0.111122
$866$	0	0
$867$	−2.27832e9	−0.118726
$868$	0	0
$869$	4.73038e10	2.44527
$870$	0	0
$871$	2.13043e10	1.09245
$872$	0	0
$873$	−3.08934e10	−1.57151
$874$	0	0
$875$	3.24482e9	0.163743
$876$	0	0
$877$	−7.53810e9	−0.377366	−0.188683	−	0.982038i	$-0.560422\pi$
$877$	−7.53810e9	−0.377366	−0.188683	+	0.982038i	$0.560422\pi$
$878$	0	0
$879$	7.98857e9	0.396742
$880$	0	0
$881$	1.42676e10	0.702967	0.351483	−	0.936194i	$-0.385677\pi$
$881$	1.42676e10	0.702967	0.351483	+	0.936194i	$0.385677\pi$
$882$	0	0
$883$	−3.13182e10	−1.53085	−0.765427	−	0.643523i	$-0.777472\pi$
$883$	−3.13182e10	−1.53085	−0.765427	+	0.643523i	$0.777472\pi$
$884$	0	0
$885$	−1.09365e9	−0.0530368
$886$	0	0
$887$	−1.11980e10	−0.538775	−0.269387	−	0.963032i	$-0.586821\pi$
$887$	−1.11980e10	−0.538775	−0.269387	+	0.963032i	$0.586821\pi$
$888$	0	0
$889$	−1.69751e10	−0.810321
$890$	0	0
$891$	−1.60473e10	−0.760027
$892$	0	0
$893$	3.73890e10	1.75697
$894$	0	0
$895$	−3.28024e8	−0.0152941
$896$	0	0
$897$	−9.97829e9	−0.461618
$898$	0	0
$899$	−3.82694e9	−0.175668
$900$	0	0
$901$	−7.81677e9	−0.356033
$902$	0	0
$903$	−1.69828e8	−0.00767541
$904$	0	0
$905$	−7.43822e9	−0.333579
$906$	0	0
$907$	1.88080e10	0.836982	0.418491	−	0.908221i	$-0.362559\pi$
$907$	1.88080e10	0.836982	0.418491	+	0.908221i	$0.362559\pi$
$908$	0	0
$909$	1.29634e10	0.572461
$910$	0	0
$911$	−4.32986e10	−1.89740	−0.948702	−	0.316173i	$-0.897602\pi$
$911$	−4.32986e10	−1.89740	−0.948702	+	0.316173i	$0.897602\pi$
$912$	0	0
$913$	3.10364e10	1.34966
$914$	0	0
$915$	4.96817e9	0.214399
$916$	0	0
$917$	1.98231e10	0.848944
$918$	0	0
$919$	−2.33634e10	−0.992962	−0.496481	−	0.868048i	$-0.665375\pi$
$919$	−2.33634e10	−0.992962	−0.496481	+	0.868048i	$0.665375\pi$
$920$	0	0
$921$	6.94326e9	0.292856
$922$	0	0
$923$	−2.66229e10	−1.11442
$924$	0	0
$925$	−9.01305e9	−0.374434
$926$	0	0
$927$	−1.53192e10	−0.631623
$928$	0	0
$929$	3.22862e10	1.32118	0.660589	−	0.750748i	$-0.270307\pi$
$929$	3.22862e10	1.32118	0.660589	+	0.750748i	$0.270307\pi$
$930$	0	0
$931$	−6.94432e10	−2.82037
$932$	0	0
$933$	−1.12696e9	−0.0454278
$934$	0	0
$935$	1.17741e10	0.471071
$936$	0	0
$937$	−2.32299e10	−0.922483	−0.461242	−	0.887275i	$-0.652596\pi$
$937$	−2.32299e10	−0.922483	−0.461242	+	0.887275i	$0.652596\pi$
$938$	0	0
$939$	−7.31132e9	−0.288181
$940$	0	0
$941$	−6.94349e9	−0.271653	−0.135826	−	0.990733i	$-0.543369\pi$
$941$	−6.94349e9	−0.271653	−0.135826	+	0.990733i	$0.543369\pi$
$942$	0	0
$943$	−2.76622e10	−1.07423
$944$	0	0
$945$	−1.48294e10	−0.571625
$946$	0	0
$947$	1.30068e9	0.0497675	0.0248838	−	0.999690i	$-0.492078\pi$
$947$	1.30068e9	0.0497675	0.0248838	+	0.999690i	$0.492078\pi$
$948$	0	0
$949$	−2.90747e10	−1.10429
$950$	0	0
$951$	3.62324e9	0.136605
$952$	0	0
$953$	−3.12573e10	−1.16984	−0.584920	−	0.811091i	$-0.698874\pi$
$953$	−3.12573e10	−1.16984	−0.584920	+	0.811091i	$0.698874\pi$
$954$	0	0
$955$	4.04189e9	0.150167
$956$	0	0
$957$	−1.94835e10	−0.718581
$958$	0	0
$959$	3.57747e10	1.30982
$960$	0	0
$961$	−2.71363e10	−0.986322
$962$	0	0
$963$	2.23644e10	0.806984
$964$	0	0
$965$	7.80129e8	0.0279461
$966$	0	0
$967$	−4.26046e10	−1.51518	−0.757590	−	0.652731i	$-0.773623\pi$
$967$	−4.26046e10	−1.51518	−0.757590	+	0.652731i	$0.773623\pi$
$968$	0	0
$969$	1.05514e10	0.372543
$970$	0	0
$971$	4.71232e10	1.65184	0.825919	−	0.563789i	$-0.190657\pi$
$971$	4.71232e10	1.65184	0.825919	+	0.563789i	$0.190657\pi$
$972$	0	0
$973$	3.20392e10	1.11503
$974$	0	0
$975$	−3.02755e9	−0.104610
$976$	0	0
$977$	−1.94783e10	−0.668220	−0.334110	−	0.942534i	$-0.608436\pi$
$977$	−1.94783e10	−0.668220	−0.334110	+	0.942534i	$0.608436\pi$
$978$	0	0
$979$	3.35756e10	1.14363
$980$	0	0
$981$	4.96771e9	0.168002
$982$	0	0
$983$	−1.93425e10	−0.649495	−0.324747	−	0.945801i	$-0.605279\pi$
$983$	−1.93425e10	−0.649495	−0.324747	+	0.945801i	$0.605279\pi$
$984$	0	0
$985$	−2.04723e10	−0.682560
$986$	0	0
$987$	−3.04258e10	−1.00724
$988$	0	0
$989$	−2.99704e8	−0.00985158
$990$	0	0
$991$	−5.22551e10	−1.70558	−0.852788	−	0.522258i	$-0.825090\pi$
$991$	−5.22551e10	−1.70558	−0.852788	+	0.522258i	$0.825090\pi$
$992$	0	0
$993$	−4.40963e9	−0.142916
$994$	0	0
$995$	−7.63048e9	−0.245568
$996$	0	0
$997$	−5.87982e10	−1.87902	−0.939509	−	0.342523i	$-0.888719\pi$
$997$	−5.87982e10	−1.87902	−0.939509	+	0.342523i	$0.888719\pi$
$998$	0	0
$999$	4.11911e10	1.30715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.8.a.e.1.2	yes	2
4.3	odd	2		160.8.a.a.1.1	✓	2
8.3	odd	2		320.8.a.r.1.2		2
8.5	even	2		320.8.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.8.a.a.1.1	✓	2	4.3	odd	2
160.8.a.e.1.2	yes	2	1.1	even	1	trivial
320.8.a.n.1.1		2	8.5	even	2
320.8.a.r.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

\(f(q)\)	\(=\)	\(q+17.5647 q^{3} +125.000 q^{5} +1661.35 q^{7} -1878.48 q^{9} -5622.79 q^{11} -11031.4 q^{13} +2195.58 q^{15} -16752.0 q^{17} -35859.5 q^{19} +29181.0 q^{21} +51497.4 q^{23} +15625.0 q^{25} -71408.8 q^{27} +197277. q^{29} -19398.8 q^{31} -98762.3 q^{33} +207669. q^{35} -576835. q^{37} -193763. q^{39} -537158. q^{41} -5819.80 q^{43} -234810. q^{45} -1.04266e6 q^{47} +1.93654e6 q^{49} -294243. q^{51} +466618. q^{53} -702848. q^{55} -629859. q^{57} -498114. q^{59} +2.26280e6 q^{61} -3.12082e6 q^{63} -1.37893e6 q^{65} -1.93123e6 q^{67} +904534. q^{69} +2.41337e6 q^{71} +2.63563e6 q^{73} +274448. q^{75} -9.34141e6 q^{77} -8.41287e6 q^{79} +2.85397e6 q^{81} -5.51975e6 q^{83} -2.09400e6 q^{85} +3.46510e6 q^{87} -5.97135e6 q^{89} -1.83270e7 q^{91} -340734. q^{93} -4.48243e6 q^{95} +1.64459e7 q^{97} +1.05623e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{3} + 250 q^{5} + 1288 q^{7} - 3974 q^{9} - 2944 q^{11} + 2028 q^{13} + 1000 q^{15} - 27644 q^{17} - 53488 q^{19} + 32752 q^{21} - 42120 q^{23} + 31250 q^{25} - 30448 q^{27} + 143228 q^{29} + 255664 q^{31}+ \cdots + 4948864 q^{99}+O(q^{100}) \) 2 * q + 8 * q^3 + 250 * q^5 + 1288 * q^7 - 3974 * q^9 - 2944 * q^11 + 2028 * q^13 + 1000 * q^15 - 27644 * q^17 - 53488 * q^19 + 32752 * q^21 - 42120 * q^23 + 31250 * q^25 - 30448 * q^27 + 143228 * q^29 + 255664 * q^31 - 124384 * q^33 + 161000 * q^35 - 324164 * q^37 - 318672 * q^39 - 257180 * q^41 - 594296 * q^43 - 496750 * q^45 - 2160920 * q^47 + 1252386 * q^49 - 190064 * q^51 + 849244 * q^53 - 368000 * q^55 - 461248 * q^57 - 2244720 * q^59 + 116332 * q^61 - 2338456 * q^63 + 253500 * q^65 - 4929384 * q^67 + 1799952 * q^69 - 3456432 * q^71 + 1393940 * q^73 + 125000 * q^75 - 10341536 * q^77 - 11340512 * q^79 + 7045090 * q^81 - 7882552 * q^83 - 3455500 * q^85 + 3982064 * q^87 - 6735180 * q^89 - 23202768 * q^91 - 2971616 * q^93 - 6686000 * q^95 + 23786180 * q^97 + 4948864 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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